# Practice Applying Rotational Motion Formulas

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• 0:04 Rotational Motion
• 1:04 Examples
• 6:18 Lesson Summary

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Lesson Transcript
Instructor: Michael Blosser

Michael has a Masters in Physics and a Masters in International Development. He has over 5 years of teaching experience, teaching Physics, Math, and English classes.

This lesson introduces the reader to rotational motion and gives examples of solving rotational motion problems such as calculating the angular velocity of an object, the rotational motion of a cylinder, and the angular kinetic energy of an object.

## Rotational Motion

Rotational motion is when an object makes a circular movement, rotating around a common axis or point of rotation. Rotational motion is important in physics as there are numerous objects that have circular shapes and rotate, as well as objects that don't always move in straight lines and instead follow curved paths and therefore have angular velocity and angular acceleration.

The angular velocity, Ï‰, of a circular object can be written in relation to its linear velocity v. This gives us the equation of:

Another important aspect of the rotational motion of a circular object is the rotational kinetic energy of an object. This equation is:

I is the moment of inertia of the object. The moment of inertia of the object depends on the shape of the object, the mass distribution of the object, and the rotation axis of each object.

## Examples

Let's go over some examples: Calculate the angular velocity of a tire of a car that has a diameter of 50 cm and is going a velocity of 25 km/hour.

We first need to determine which equation to use. Since we're being asked for the angular velocity and were given the linear velocity, we should use the angular velocity formula in relation to its linear velocity. First, though, we need to convert the velocity of 25 km/hour to a velocity in meters/second.

Now, we can plug in our value of v and solve for the angular velocity Ï‰, giving us:

This gives us a final angular velocity Ï‰ of 27.76 radians/second.

Let's try another problem: Calculate the kinetic energy of a cylinder that has a radius of 5 cm, which is moving at 15 radians/second with a 5 kg mass uniformly distributed throughout the cylinder.

Another important rotational motion formula is the rotational kinetic energy of an object. The rotational kinetic energy is dependent on the object's angular velocity, moment of inertia, and radius and is calculated using this equation:

The moment of inertia, I, of each object is different and is dependent on an object's shape, mass distribution, and point of rotation. The question asks us to calculate the kinetic energy of a cylinder with a uniformly distributed mass. Through calculus, we find out that the moment of inertia of a cylinder with a uniform mass distribution can be calculated with the following equation:

Icylinder with uniform mass = 1/2 mr2

Therefore, plugging this into our KE equation and then plugging in our given variables, we get:

This gives us a rotational kinetic energy of 0.703 Joules.

What would the kinetic energy of the cylinder be if all its mass was concentrated at the center of the cylinder? Solving for cylinder with mass concentrated at center, we need to use the moment of inertia of:

Icylinder with mass at center = 0.1 mr2

Therefore, plugging this into our kinetic energy equation and then plugging in our given variables gives us:

This gives us a rotational kinetic energy of 0.141 Joules.

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